(2) Let us deal with your second suggestion first. I agree with you that everybody says the best solution is to short the transistor's CB and made it a diode then use the 2.2 mV and follow, say what you suggest, use a current source to apply constant current to the diode and the problem is almost solved.

But our customer might not be very happy to what you suggest for two reasons, (a) He asks for the transistor question, but you twisted it to a diode question,

Your suggestion is worse than the following: "Throw away the 2N222 and found a diode, just any diode, and search google for "2.2mV", "Diode, temperature, sensor". The point is that the OP might have spent hours on biasing the transistor, perhaps in the wrong way as I biased it as a switching transistor, in the saturated mode.

If he shows me a photo or a schematic, I would repeat his situation and found that I actually did not know how to use 2N2222 as an amplifier. Then by trial and errors I found that I should bias it in the current amplification region and solved the problem. Now the OP might need to only change the Rb of Rc and solved the problem, and understand and learn how to use the transistor in two ways, as a switch and as a small signal amplifier. That is what I said "PBL", / to continue, ...

Even I have no necessity for now, I usually browse a couple of forums and if newbies have any necessity, and I would try to invent or innovate to let the OP become sort of a spiritual rich guy with no necessity, because all problems solved, ... :(

(1) Let us go back to your first suggestion of "Let us don't do any analysis, and just start with the spec (temperature range and accuracy)" which does make me laugh, because it is just non engineering, / to continue, ...